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On nonlinear viscoelastic deformations - a reappraisal of Fung's quasilinear viscoelastic model
No full textThis article offers a reappraisal of Fung's method for quasilinear viscoelasticity. It is shown that a number of negative features exhibited in other works, commonly attributed to the Fung approach, are merely a consequence of the way it has been applied. The approach outlined herein is shown to yield improved behaviour, and offers a straightforward scheme for solving a wide range of models.
Results from the new model are contrasted with those in the literature for the case of uniaxial elongation of a bar: for an imposed stretch of an incompressible bar, and for an imposed load. In the last case, a numerical solution to a Volterra integral equation is required to obtain the results. This is achieved by a high order discretisation scheme. Finally, the stretch of a compressible viscoelastic bar is determined for two distinct materials: Horgan-Murphy and Gent
Testing matrix function algorithms using identities
No full textAlgorithms for computing matrix functions are typically tested by comparing the forward error with the product of the condition number and the unit roundoff. The forward error is computed with the aid of a reference solution, typically computed at high precision. An alternative approach is to use functional identities such as the ``round trip tests'' and , as are currently employed in a SciPy test module.
We show how a linearized perturbation analysis for a functional identity allows the determination of a maximum residual consistent with backward stability of the constituent matrix function evaluations. Comparison of this maximum residual with a computed residual provides a necessary test for backward stability. We also show how the actual linearized backward error for these relations can be computed. Our approach makes use of Fr\'echet derivatives and estimates of their norms. Numerical experiments show that the proposed approaches are able both to detect instability and to confirm stability
Implementing QR Factorization Updating Algorithms on GPUs
No full textLinear least squares problems are commonly solved by QR factorization. When multiple solutions need to be computed with only minor changes in the underlying data, knowledge of the difference between the old data set and the new can be used to update an existing factorization at reduced computational cost. We investigate the viability of implementing QR updating algorithms on GPUs and demonstrate that GPU-based updating for removing columns achieves speed-ups of up to 13.5x compared with full GPU QR factorization. We characterize the conditions under which other types of updates also achieve speed-ups
Black box, white arrow
No full textThe present paper proposes a new and systematic approach to the so-called black box group methods in computational group theory. Instead of a single black box, we consider categories of black boxes and their morphisms. This makes new classes of black box problems accessible. For example, we can enrich black box groups by actions of outer automorphisms. As an example of application of this technique, we construct Frobenius maps on black box groups of untwisted Lie type in odd characteristic (Section
6) and inverse-transpose automorphisms on black box groups encrypting (P)SL_n(F_q).
One of the advantages of our approach is that it allows us to work in black box groups over finite fields of big characteristic. Another advantage is the explanatory power of our methods; as an example, we explain Kantor's
and Kassabov's construction of an involution in black box groups encrypting SL_2(2^n).
Due to the nature of our work we also have to discuss a few methodological issues of the black box group theory
On Pi-Product Involution Graphs in Symmetric Groups
Get PDFSuppose that G is a group, X a subset of G and pi a set of natural numbers. The pi-product graph pi(G,X) has X as its vertex set and distinct vertices are joined by an edge if the order of their product is in pi. If X is a set of involutions, then pi(G,X) is called a pi-product involution graph. In this paper we study the connectivity and diameters of pi(G,X) when G is a finite symmetric group and X is a G-conjugacy class of involutions
An optimized and scalable eigensolver for sequences of eigenvalue problems
Get PDFIn many scientific applications the solution of non-linear differential equations are obtained
through the set-up and solution of a number of successive eigenproblems. These eigenproblems
can be regarded as a sequence whenever the solution of one problem fosters the initialization
of the next. In addition, in some eigenproblem sequences there is a connection between the
solutions of adjacent eigenproblems. Whenever it is possible to unravel the existence of such a
connection, the eigenproblem sequence is said to be correlated. When facing with a sequence
of correlated eigenproblems the current strategy amounts to solving each eigenproblem in isolation.
We propose a alternative approach which exploits such correlation through the use of an
eigensolver based on subspace iteration and accelerated with Chebyshev polynomials (ChFSI).
The resulting eigensolver is optimized by minimizing the number of matrix-vector multiplications
and parallelized using the Elemental library framework. Numerical results show that
ChFSI achieves excellent scalability and is competitive with current dense linear algebra parallel
eigensolvers
Diffraction tomography of strain
No full textWe consider the tomographic recovery of strain from x-ray and neutron diffraction data of a polycrystalline material. We show that the distribu- tion of strain transverse to a ray cannot be deduced from one diffraction pattern accumulated along that path, but a certain moment of that data corresponds to the transverse ray transform of the strain tensor and so may be recovered by inverting that transform given sufficient data. We give an example of sufficient data characterised by the data measured from rotations about six direction that do not lie on a projective conic. In addition we give an inversion formula for complete data for the transverse ray transform. We also show that a certain type of Bragg edge data which has been suggested for strain tomography with polychromatic data, can in fact only determine the change in the external dimension of a solid specimen and is otherwise unaffected by the distribution of strain along the ray
Diffraction tomography of strain
Get PDFWe consider whether it is possible to recover the three dimensional strain field tomographically from neutron and X-ray diffraction data for polycrystalline materials. We show that the distribution of strain transverse to a ray cannot be deduced from one diffraction pattern accumulated along that path, but that a certain moment of that data
corresponds to the transverse ray transform of the strain tensor and so may be recovered by inverting that transform given sufficient data. We show that the whole strain tensor can be reconstructed from diffraction data measured using rotations about six directions that do not lie on a projective conic. In addition we give an inversion formula for complete data for the transverse ray transform. We also show that Bragg edge transmission data, which has been suggested for strain tomography with polychromatic data, cannot provide the strain distribution within the material but only the average along the ray path
New Algorithms for Computing the Matrix Sine and Cosine Separately or Simultaneously
Get PDFSeveral existing algorithms for computing the matrix cosine employ polynomial or rational approximations combined with scaling and use of a double angle formula. Their derivations are based on forward error bounds. We derive new algorithms for computing the matrix cosine, the matrix sine, and both simultaneously, that are backward stable in exact arithmetic and behave in a forward stable manner in floating point arithmetic. Our new algorithms employ both Pad\'e approximants of and new rational approximants to and obtained from Pad\'e approximants to . The amount of scaling and the degree of the approximants are chosen to minimize the computational cost subject to backward stability in exact arithmetic. Numerical experiments show that the new algorithms have backward and forward errors that rival or surpass those of existing algorithms and are particularly favorable for triangular matrices
Estimating the Condition Number of f(A)b
Get PDFNew algorithms are developed for estimating the condition number of , where is a matrix and is a vector. The condition number estimation algorithms for already available in the literature require the explicit computation of matrix functions and their Fr\'{e}chet derivatives and are therefore unsuitable for the large, sparse typically encountered in problems. The algorithms we propose here use only
matrix-vector multiplications. They are based on a modified version of the power iteration for estimating the norm of the Fr\'{e}chet derivative of a matrix function, and work in conjunction with any existing algorithm for computing . The number of matrix-vector multiplications required to estimate the condition number is proportional to the square of the number of matrix-vector multiplications required by the underlying
algorithm. We develop a specific version of our algorithm for estimating the
condition number of , based on the algorithm of Al-Mohy and Higham [SIAM J. Matrix Anal. Appl., 30(4):1639--1657, 2009].
Numerical experiments demonstrate that our condition estimates are reliable and of reasonable cost